32 Basic shapes of geometry Ch. 2
(iv) Each of the sides[B,C],[C,A],[A,B]is a subset of[A,B,C].Proof. (i) As∩is commutative,H 1 ∩H 3 ∩H 5 is independent of the order of
H 1 ,H 3 ,H 5.
(ii) The vertexAis inH 1 by definition. It is also in the edge of each ofH 3 and
H 5 , so by 2.2.3 it is in each of these closed half-planes. The verticesBandCare
treated similarly.
(iii) By definition of an intersection,PandQare in each ofH 1 ,H 3 ,H 5 .By
2.2.3,[P,Q]is a subset of each of these closed half-planes, and so it is a subset of
their intersection.
(iv) This follows from parts (ii) and (iii) of the present result.
2.4.3 Pasch’s property, 1882
Pasch’s property.If a line cuts one side of a triangle, not at a vertex, then it will
either pass through the opposite vertex, or cut one of the other two sides.
Proof.Let[A,B,C]be the triangle andla line which cuts the side[A,B]at a point
which is not a vertex. IfC∈lwe have the first conclusion. Otherwise suppose thatl
does not cut[B,C].ThenAandBare on different sides ofl,butBandCare on the
samesideofl. It follows thatAandCare on different sides ifl,sobyA 3 (iii) a point
of[A,C]lies onl.
2.4.4 Convex quadrilaterals
Definition.LetA,B,C,Dbe four points inΠ, no three of which are collinear, and
such that[A,C]∩[B,D]=0. Let/ H 1 be the closed half-plane with edgeABin which
Dlies,H 3 the closed half-plane with edgeBCin whichAlies,H 5 the closed half-
plane with edgeCDin whichBlies, andH 7 the closed half-plane with edgeDAin
whichClies. Then the intersectionH 1 ∩H 3 ∩H 5 ∩H 7 of these four half-planes is
called aconvex quadrilateral, and we denote it by[A,B,C,D]. As each of the closed
half-planes is a convex set this intersection is also a convex set.
Each of the four pointsA,B,C,Dis called avertex; the segments[A,B],[B,C],
[C,D],[D,A]are called thesides,andAB,BC,CD,DAare called theside-lines;the
union of the sides[A,B]∪[B,C]∪[C,D]∪[D,A]is called theperimeter. The segments
[A,C],[B,D]are called thediagonals,andAC,BDthediagonal lines. Vertices which
are the end-points of a side are calledadjacentwhile vertices which are the end-
points of a diagonal are calledopposite; thusAandBare adjacent as[A,B]is a side,
andAandCare opposite as[A,C]is a diagonal. Sides which have a vertex in common
are said to beadjacentwhile sides which do not have a vertex in common are said to
beopposite; thus the sides[A,B],[A,D]are adjacent as the vertexAis in both, while
the sides[A,B],[C,D]are opposite as neitherCnorDis inABand so neither of them
could beAorB.